 Welcome back to this today's lecture, we have been studying statistical mechanics. More importantly, we are trying to understand many aspects of natural phenomena that we study in the name of physical chemistry and chemical physics and those things the understanding of the huge number of physical and chemical phenomena like solutions, chemical reaction dynamics, phase transitions, all these important things important both in physics, chemistry, biology, material science, physical chemistry get you started on them because it is through physical chemistry you learn about properties of solvents, it is through physical chemistry you learn chemical kinetics, you learn electrolytes. Now how does physical chemistry do that? If you reflect a little bit the physical chemistry that you studied in your undergraduate books or even in some time in advanced schools, physical chemistry introduces many many concepts. In certain sense it starts with kind of clear of gases that we have told many times then it might also takes the help of thermodynamics and we get to give free energy and emerge free energy all these different conditions and then goes into talk little bit of phase transition. So physical chemistry tells you phenomena and tells you to a certain phenological explanations of that and to be specific to the today's lecture let me take the case of binary mixtures. A mixture of two species A and B in the language of physical chemistry I will talk one as the solvent and one as the solute. The other alternative language more popular in organic chemistry is solvent and co-solvent. Solvent and solute has the disadvantage because it always assumes that one is in small quantity which is the solute but many times we have 50-50% mixture of two species B and B. Now we know from physical chemistry that a binary mixture of two species like the very common binary mixtures is water and ethanol and they show enormous number of range of properties. Binary mixtures are very common because these are much of our common solvents are a binary mixtures because many things are not soluble in water. So you add a little co-solvent like add a little ethanol and become and water is a fantastic medium for reactions and it has lot of properties like the polarity. So another solvent if we can mix it together and tune the properties then we get a huge amount of diverse range of properties by tuning by varying the composition. But how do you know about the interactions between them? How do you know how water and ethanol interact? We might know about water we might be able to construct a force field that means how two water molecules interact there is a huge amount of work that has gone into it or even ethanol some amount of work has gone into it. Water and ethanol still is somewhat less thing but is very important. So the water starting into a mixture by the characteristics of binary mixtures starts in the following way. We construct, we will study a property P and we have added the mole fraction. We go on adding increasing the solute little by little and then start the property. What are the properties we are going to study? We can start the properties for example what is the volume or excess volume and I will define what is excess volume? What is the volume of the mixture? We can add constant pressure sometime when you add a co-solving the volume of the two decreases then what it should be if they are ideally mixed we talk of viscosity we can talk of the dynamical properties like diffusion. So the range of property is that an experimentalist measures and you will study in that experiment is binary mixtures. In India it was a very common and very popular research of the study of binary mixtures. I know many many people who studied whole their life on binary mixtures and it quite significant work. Another method of doing experimental study of binary mixtures is through sound attenuation you send a sound wave and you find out how the sound get attenuated how sound get absorbed. There are a host of thermodynamics and dynamic experimentalist tool that we bring in to understand these binary mixtures. So what I will do now in the in this lecture today we will talk of binary mixtures, motivate it and tell you how this very important class of physical systems very important for physical biological science how we could begin and understanding of the physical properties of these things. And this is done by as I said the statistical mechanics and this is do we describe in this book that we have written and this is in chapter 19 or something chapter 26 in this book that we have done the discussed the subject in quite great detail and we will follow this book closely but there will be some simplifications that will okay. So I just recap what I said that binary mixtures constitute wide and highly useful class of solvents then properties are quite different from those of the PO solvents. That means I have A properties of A and properties of B when I mix them A and B has properties which are very different from that of A then something which you have studied at length probably even in schools but certainly in your physical chemistry is Raoult's law Raoult's law says that the properties of a property P is X1 P1 X is the mole fraction. So if there are two species 1 and 2 if I mix them then property P of the mixture becomes mole fraction of species 1 multiplied by the property P1 then for example I can take of viscosity then viscosity would be written as X1 N1 plus X2 N2 then the composition dependence for mole fraction ideal Raoult's law gives you a certain dependence on mole fraction which is here but you find a very large deviation from that. Now many properties are obtained by using the Ising model as we discussed in a previous lecture that how Ising model now the binary mixture can be mapped quite well into the Ising model it was really done in the more appropriate in the context of the binary alloy. The reason it is more is the binary alloy because I can put a binary alloy in a lattice like brass, beta brass that is copper inch, this is a lattice space center to the lattice and then I can it is easy to map a liquid in a desoldered systems and in the next lecture or two will be will be discussing property the structure of liquids that is different that means structure of liquids is quite different from that of a binary alloy or that because here we have more use of statistical mechanics because properties are much more statistical that means we have to talk in terms of probabilities. So let us get started then so some simple eye opening kind of things that you have one species A by the red balls and then blue by the B and we mix them and one thing that you realize immediately that A and B can interact differently. So if I have an interaction potential and they can have diameters differently. So the diameters let us see these diameters is sigma A and these diameters is sigma B then A interact with the interaction potential A, A, B interact with the interaction of B, B. So what do I mean by that A, B and A, B I have what in back of my mind essentially a interaction potential you are and I have this kind of potential inner zones then it is the depth that I am talking of course harsher part of the potential is important but harsher part of the potential makes its presence fail in giving you the structure of the liquid and but the thermodynamics the internal energy enthalpy many things which are property of the which are manifestation of property of the structure is really very interesting role that this part and this part plays you know that means one part give you the structure then the thermodynamics that structure of course the thermodynamics and dynamics but on that structure you have a liquid because you have internal interactions is epsilon. So these two play very interesting role together and that I think I want you to appreciate. So now vector outlaw as I said outlaw is a law which is called ideal solution. So ideal solution then ideal will be p ideal or p rows p r you can call it p r p rows is just what I wrote down however it is a very remarkable many, many solvents so very minor remarkable deviation from this ideal if it are ideal then these should be the line these should be the line of the straight line these should be the line but what do you find the all kinds of behaviors the real numbers like here carbon tetrachloride erythyl acidate here you have a cyclohexane which is very quite inner solvent inner solvent and then ethanol then that has a mar deviations this is a positive deviation of viscosity yes and this is a negative deviation of viscosity and here in the methanol and toluene these are fairly common solvents you know we all chemists all know about it that carbon tetrachloride erythyl acidate cyclohexane ethanol bethanol and toluene but look at the kind of behavior they show one is a positive deviation and there is a negative deviation and there it is negative now this kind of behavior has given rise to certain nomenclature and this nomenclature is the one that we will go into now and then we will try to do stress chemical mechanism so purpose of this chapter is partly to try to tell you how we can understand these things this is still very much an active area of research and lot of papers are being published and that one of the reason that such such sudden interest in the last 10-15 years in this area is the use of complicated simulations these as you can see carbon tetrachloride erythyl acidate or cyclohexane they are common solvents but they are complex molecules they are not spheres they are not like the spheres we just do in the previous slides so they are more complex molecules and so the interactions are very complex so our analytical work towards this has been as a slow our progress in analytical work and so it needed to for computers to become really important and then of course in order to get the computer going I need the interaction potential which is the force field and that force field also has been developed or being developed for all these systems and that is a very rigorous and very respectable way right now the developing of force field is a for force field development of force field for allows us to simulate the system is a very respectable branch of work in physical chemistry and so on the way along I will tell you what are the fields that one is working so that you basically know what is the current status so then physical chemistry attains statistical mechanics attains so this is what physical chemistry handed us from under graduate these these these and now physical chemistry statistical mechanics rather wants to provide an explanation for these things and how do we do now we start with an interaction potential and we find that this kind of thing which is we where the viscosity goes up that means the structure becomes more rigid that is called structure forming here viscosity become less and we call this structure breaking and here there is a crossover from structure breaking in the negative part to structure making so when a structure breaking the structures they will be and we like each other and structure making the weakest structure which is coherent and that becomes more rigid viscosity goes up and when there is structure breaking there they are kind of stay away from each other and there are there is always an entropic component but there is an entropic component but and they interact with each other in in these two cases to give this kind of a bizarre and wide and different behavior okay so how then we go in statistical mechanics everything starts with the potential now we have the interaction potential such that we called a structure forming liquid stronger interaction between a and so now a b like a and b like each other more than a a like each other b b like each other similarly there is a model 2 which is structure breaking in this case we assume either by note a by convention a a like each other more than b b like each other but most importantly both of this interaction potential that I continuously will draw this thing is more than a and b that mean a and b we do not like each other or they like they might like each other but they like less than b likes b and a likes a this is structure breaking and structure breaking is the one that gave this viscosity if I wrote viscosity against composition this is the Ralph's law this goes like that the negative deviation is viscosity and will be positive deviation excess volume if I plot the excess volume excess volume and excess viscosity go in the other way because when the volume increases excess volume is positive then volume increases on addition of two solvents then the viscosity goes down because the molecules kind of expand they kind of go away from each other okay and that is the structure breaking state they come closer together so the excess volume goes down becomes negative and if viscosity goes up okay so our force field then we write a we write in a we are doing the simplest possible thing now then we will go into make it much more rigorous but at the level of starting we are starting to build a theory starting to build an understanding based on certain mechanics so we start into force field the force field is the reandoms interaction potential and our old friends for epsilon but now a and b like a and b is for when you have this for u ij is very u ij so when I have a u a then that would be for epsilon a a and the rest of the things sigma a sigma b so this ij is important and this should actually be lower okay so again to recap the whole thing if this is positive this is a taking a model like that then did some theoretical work both computer simulations and some sophisticated statistical mechanical theory and you can see now that you can generate by this kind of a model that we described here these kind of model one can indeed describe so this would be the Raoult's law you have the positive deviation and negative deviation so the model that I am so this is structure breaking and the structure making this so what I am telling you in victory any can one can do and I will describe how does one do that in a in a next half an hour so now there are some beautiful phenomena we studied little bit of it is called and it is given in the book in the stat mac stat mac book by CRC that we done now there is a very interesting phenomena that a and b that I was I was just going to address if a and b they do not like each other how come they still together in a liquid well if I ask you the question I think some of you already know and we will be able to give the answer they stay each other because what is the fact what is the reason that they stay together because the entropy when you form a binary mixture you get a large entropy that is entropy of mixing so that entropy of mixing contributes to making free energy negative so the mixture it makes but what happened if I suddenly cool the temperature if I suddenly cool the temperature then entropy contribution become this then they cannot stay together then they face separate so the structure breaking liquid where a and b they do not like each other but still are together in high temperature they lower temperature they face separate and space separate through a beautiful pattern formation which is we see in nature all the time in volcanic rocks and in all the other systems and these are those kind of the beautiful structures that I am showing you here is this that so so a and b are two different spaces so the a sorry red and blue are two different spaces if this is or no this is our a and this is our b then when you lower temperature then they become beautifully face separate this is another very interesting aspect of binary mixtures which is very popular in in binary alloys and very popular in material science okay next we go and we now describe two cases where which are very very common I am still in the process of entering the introduce the students to this beautiful beautiful world of binary mixtures one is water and dimethylsulfoxide the two popular two popular and useful highly useful highly useful in chemistry and biology is water dimethylsulfoxide DMSO and water ethanol so DMSO let me draw the DMSO then DMSO is is a sulfur oxygen CH3 CH3 this is the dimethylsulfoxide and ethanol you know CH3 CH2 OH so the and then I have water so everywhere there is water here these are water molecules now you notice something very interesting in the dimethylsulfoxide oxygen is negatively charged sulfur is positively charged and me a methane group nearly spherical has very little charge so water molecules just surround this oxygen and water molecules kind of go away from you know they kind of form a cage structure but they little distance away because they want don't want to disrupt the hydrogen bond ethanol again has these two parts which are not very polar so they don't form hydrogen bond but these guys from hydrogen bond like oxygen oxygen form hydrogen bond to hydrogen bond hydrogen from one hydrogen bond so there is a show these kind of molecules are particularly good because in these molecule a and b together so I have a molecule where a and b together I would say this is my structure making thing and this is my structure breaking thing if a structure making structure making then together with respect to what with respect to water molecules so this is called hydrophilic group this is called hydrophobic group and these kind of solvents which are very important are that called apophilic solvents okay we will now go doing and when you do that then the kind of spinality composition and in dimethylsulfoxide you see this kind of beautiful phase separation you know first they are together and so the they are in a small composition they kind of form they kind of form micro micro droplets like this droplets showing here but when I increase the composition this will be what may be 10% and this would be what would be by 20% that they would here islands are connected and lot of many beautiful properties of water so very many beautiful properties because of these two these two phenomena so what I am trying to tell you now is that we are going to do a quantitative theory we are going to explain a quantitative theory but you should never you should never fail to appreciate that there is certain understanding which is coming from physical chemistry the understanding in terms of structure making structure breaking is a very powerful paradigm then hydrophobic and hydrophilic this is again a very powerful concept so what station mechanics does takes this powerful concept and terminology and picture from physical chemistry and then kind of put them into a quantitative predict theory so these beautiful language that we use and the concept they are not predict they do not give you numbers they give you some understanding but they do not they are not quantitative they are qualitative so but now station mechanics what gives you a quantitative they give you how to add okay now let them get going now as I am telling you that we want to a and b interaction potential a so I have some idea of the interaction between a and a I have some idea of interaction between b and b but I do not have any idea of the understanding between a and b how do I find now interaction and in a and b now remember go back again and again I told you something that the force field before how do you get the force field the first force field was done after the expression of second virial coefficient b2 in terms of interaction potential you are remember the expression of b2 is it is a integration over dr r square e to the power minus u r by k b t minus 1 so I have a temperature dependence of experiment later mine temperature dependence of the second virial coefficient so now I can put it into a form like a little form and now I can fit I can evaluate away these if I have this temperature then I should have this second virial coefficient and I already have measured second virial coefficient so I have a non-linear field and I get the properties the epsilon and sigma of the linear potential that is this is called force field then I can do many other things now for a and b we have host of properties we have diffusion constant we have the equation of state we have second virial coefficient so we have a control over sigma and epsilon the force field parameters of the internal but how about binary mixtures how about a and b that is much more complicated because if I now want to say okay let me play the game of second virial coefficient dot but then problem is that when I do that I have not just have to do with a and b I have to deal with a and b b also so then the basic idea is that divide and conquer we get a and a we get b b b then I want to make a and b but then I need a phenomena certain certain particular experiments which is sensitive to interaction between a and b because I already have two other interactions hanging around and confusing and that is the osmotic pressure osmotic pressure gives you this your in road into interaction between a and b the reason their osmotic pressure is so important that the osmotic pressure has been taught in such great detail in other kind of physical chemistry it is a very big thing in biology in if you have biophysical chemistry then osmotic pressure is extremely important and we will explain now that how that goes and one of the major thing of this particular lecture of statistical mechanics is bringing home that particular point that statistical mechanics gives you an expression for osmotic pressure and then you can from there it essentially defines a secondary coefficient of the solute and from there you can find the interaction potential between a and b is very important I need to know how a and b interact experimentally and okay that is let us get started then so osmotic pressure as defined here that you say when I put a and b together then and I put them in a semi permeable membrane then the semi permeable membrane that allows a but not does not allow b so a starts coming from one side to the other like showing here it shows from goes from a to b and how do I stop that I stop it by putting a pressure up there so these guys in pressure p but these guys in pressure p plus these in p plus pi and pi is the atmospheric pressure we know that so we put a pressure on right side where I have solute due to pressure on solution to stop the flow and that excess pressure is the osmotic pressure now when I stop the flow then the systems are in equilibrium now there in chemical equilibrium there in mechanical equilibrium there in thermodynamic equilibrium chemical potential of the two are the same which is in here so chemical potential a solv is for solvent chemical potential of the solvent molecules so it will be particular we have two chemical potential but chemical potential of the solvent molecules because it is solvent molecule that was coming from left to right same on the two sides so so the solute chemical potential of the solvent solvent is the same so chemical potential of the solvent must be the same on the two sides as is means in the presence of this solution these I know these you not pt this is the chemical potential of these guys so now from thermodynamics I can now if I have mu in the presence of small small concentration of a solute then I can write that the chemical potential is written as the that of the pure solvent so this is a interesting beast this is say at this is a bold fraction of the solvent that p plus pi is the pressure which I am talking of this this right hand side of my slide and then at temperature T then I now write it in for in terms of the pure solvent but it has to be a pressure p plus pi and then the correction term that is RTLNX solvent okay this comes from thermodynamics so now since I have this thing and this thing this thing now must be by equation 1 and equation 2 I equate these two and I get mu naught p plus pi this guy here RTLNX i is mu naught pt so this is the this is the one equation that I derived this is only one of the equations on the way we are going to derive them but remember this equation next equation we are going to do is the following okay now we have to use the Gibbs Bohm equation what you are going to do now we are going to the whole idea is if you smart enough you would be knowing from ideal gas law or from gas law that we are I whole idea is to get whole aim is to get an expression for the optimal pressure pi right that is the goal that is why I am doing all these things that is what I have written derive the equation in chemical potential so now if I do that is my star equation here I want to use it but how do I go about now now I need then next piece of the puzzle next piece of the puzzle is to find out the variation of chemical potential as a function of pressure I have done variation of chemical potential in terms of mole fraction what I am going to do now as a function of pressure right and who gives you that who gives you variation of chemical potential with pressure that is the Gibbs Bohm equation that is one of the really I equation I like and it is a very very powerful equation and that is why Gibbs Bohm equation is so so important so here is the Gibbs Bohm equation N d mu V dp sorry about the notation of P many different kind of me but it is actually P is pressure here I divide this by N and divide the space V volume V then I integrate from pressure P to pressure P plus P P plus pi pi is the optimal pressure I integrate d mu then I integrate this guy I will of course integrate this one in a minute and then I integrate I can do this integration this mu naught P plus pi t minus mu naught P t the same thing as integration of V dp because the integration of that number so I have first done just the integration and the and got this thing then they I put it equal to V dp I probably would have in other way around okay now this integration is from P to P plus pi so I would have probably written it dp V dp equal to then there is then integration mu naught P plus pi forget about P so now we are almost there so now I bring this the combined the two things together and I get V dp and now what we are going to do we are going to go back to the previous slide we have already derived that so I bring this on the side so mu naught P plus pi minus mu naught P equal to RTLN so I my equation is done so I have this now I assume V is weakly dependent on pressure the molar volume is increasingly dependent on pressure then I get the integration the V comes out this comes out and then this integration just give me pi so V pi equal to RTLN X and take I take on the right hand side so I derived this beautifully this was done by great Vantov one of the three major people who is physical chemistry one is Arrhenius the other is Ostwald another is Vantov and first Nobel Prize in chemistry went to Arrhenius second one went to Vantov third one would have gone to Ostwald but presumably story I know that Arrhenius did not like Ostwald at all so this in 2007 was Ostwald Nobel Prize 1901 was Arrhenius 1902 or 1903 Vantov I remember I was asked this question in a one of the quiz because in India we have very strange kind of factory entered exams and I was asked who got the first Nobel Prize in chemistry I thought why to be chewed my pain and then I selected Vantov but no it was Arrhenius who got the first Nobel Prize but you know what Vantov is one of the father figure of the solution phase and thought that was what was the early 20th century that was the most important because people realize that we must understand solution okay so this beautiful relation now now something even more interesting we are going to do this is RTLN X solvent now I write it as X solvent is 1 minus X solute this is here then LN 1 minus X solute is X solute minus this is the expansion logarithmic expansion and no and behold I get a beautiful so I take this term first term minus and minus plus and I get my ideal gas law this is the ideal gas law for solutions this is one of the most important thing of the binary mixture osmotic pressure is the most important thing and this equation Vantov all these things were done by Vantov so we now beginning to get an idea we are now beginning to get an idea okay okay this is how it goes that I have been able to derive the ideal gas law now if I get a deviation from the ideal gas law in terms of the concentration X solute then I will be able to get a second ideal coefficient thing like that so I will then get an idea of interaction effective interaction between B and B you know and which is very useful and also interaction between A and B so that was exactly the strategy was followed by mayor in many years later and it was also the strategy that was Vantov and Ostwald has is really all the really makes a beautiful